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62                        ANALYTICAL MECHANIC'S
It is evident from equation (1) that the horizontal component of the tensile force is constant and equals 7'«. Squaring equations (1) and (2) and adding wo get
T* = Tf + uW.                                  C{)
Thus we see that the smallest value of T corresponds to x = 0 and equals TO, while its greatest value corresponds to the greatest value of x. If I) denotes the span of the bridge then the greatest value of 7T, or the tensile force of the cable at the piers, is
In order to find the equation of the curve which the cable assumes we eliminate T between equations (I)  and   (2).
This gives
tan0 = --#.                                 (4)
Substituting -^for tan 6 and integrating we got
1    W       n   ,
^       .~.x-+c,
Z,  Jo
where c is the constant of integration.
But with the axes we have chosen, //-() when x    ()» therefore c= 0.   Thus the equation of the4 curve* is
which is the equation of a parabola* DIP OF THE CABLE. —Let // be the height of the pirn
above the lowest point of the cable.   Then for x --• ^, y    //, therefore
'-sk*                        ('•"
It is evident from the last equation that the greater the tension the less is the sag.